zigya tab
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
4 straight u squared plus 8 straight u
      
     

We have,
4u2 + 8u = 4u (u + 2)
So, the value of 4u2 + 8u is zero when 4u = 0 or u + 2 = 0
i.e., when u = 0 or u = –2
Therefore, the zeroes of 4u2 + 8u are 0 and - 2
Now,
Sum of zeroes = 0 + (-2) = fraction numerator negative 2 over denominator 1 end fraction
                                         equals fraction numerator negative left parenthesis Coefficient space of space straight u right parenthesis over denominator left parenthesis Coefficient space of space straight u squared right parenthesis end fraction
Product of zeroes = 0(-2) = 0 over 1
                            = fraction numerator Constant space term over denominator Coefficient space of space straight u squared end fraction

870 Views

The graph of y = p(x) are given in the figure below, for some polynomials p(x). Find the number of zeroes of p(x), in each case.


(i) Graph y = p(x) does not cut the x-axis at any point, so the given polynomial has no zero. (ii) Graph y = p(x) cuts the x-axis at one point, so the given polynomial has one zero. (iii) Graph y = p(x) cuts the x-axis at three points, so the given polynomial has three zeroes. (iv) Graph y = p(x) cuts the x-axis at two points, so the given polynomial has two zeroes. (v) Graph y = p(x) cuts the x-axis at four points, so the given polynomial has four zeroes. (vi) Graph y = p(x) cuts the x-axis at three points, so the given polynomial has three zeroes.
2329 Views

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
6 straight x squared minus 3 minus 7 straight x
      
     

We have,6x2–7x–3 = 6x2– 9x + 2x – 3
= 3x(2x – 3) + 1(2x – 3)
= (3x + 1) (2x – 3)
So, the value of 6x2 – 7 – 3 is zero when (3a + 1) = 0 or (2x – 3) = 0
i.e.                 <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>
Now, 
   Sum of zeroes  = <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>

                            space space space equals fraction numerator negative left parenthesis Coefficient space of space straight x right parenthesis over denominator Coefficient space of space straight x squared end fraction

Product of zeroes = <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>

                            = fraction numerator Constant space term over denominator Coefficient space of space straight x squared end fraction.
862 Views

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
      4 straight s squared minus 4 straight s plus 1
     

We have,
4s2 – 4s + 1 = 4s2 – 2s –2s + 1 = (2s–1)(2s– 1)
So, the value of 4s2 – 4s + 1 is zero, when 2s – 1 = 0 or 2s – 1 = 0
i.e.  when   straight s equals 1 half space or space straight s equals 1 half
Therefore, the zeroes of 4s2 – 4s + 1 are 1 half space and space 1 half. Now,
     Sum of zeroes =  1 half plus 1 half

                             equals 1 over 1 equals fraction numerator negative left parenthesis Coefficient space of space straight s right parenthesis over denominator Coefficient space of space straight s squared end fraction

Product of zeroes  = 1 half cross times 1 half equals 1 fourth
                              <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>


791 Views

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
     straight x squared minus 2 straight x minus 8

We have,
      space space space straight x squared minus 2 straight x minus 8 space equals space straight x squared minus 4 straight x plus 2 straight x minus 8
                              <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>
So, the value of x2 – 2x - 8 is zero when x - 4 = 0 or x + 2 = 0.
i.e., when x = 4 or x = –2.
Therefore, the zeroes of x2 – 2x – 8 are 4 and - 2. Now,
   Sum of zeroes = (4) + (-2)
                           equals 2 over 1 equals fraction numerator negative left parenthesis Coefficient space of space straight x right parenthesis over denominator Coefficient space of space straight x squared end fraction
Product of zeroes = (4) (-2)
                             equals fraction numerator negative 8 over denominator 1 end fraction equals fraction numerator Constant space term over denominator Coefficient space of space straight x squared end fraction.


931 Views